Research
A two-part mathematical study of finite-resource stochastic dynamics, irreversible energy depletion, and emergent structural hierarchy on discrete graphs. No continuum limit. No external drive. No infinite-resource assumption.
Author: Shiv Goswami, Terraflock Pvt Ltd, Bihar, India. Submitted for journal.
github.com/Shiv2071/Discrete-cascade-modelPart I
5 July 2026
A discrete-time stochastic dynamical system on finite graphs with irreversible energy depletion, transient structural hierarchy, and almost-sure finite-time absorption. Establishes three rigorous guarantees: energy monotonicity as a supermartingale, finite total activity, and almost-sure absorption in finite time.
Part II
6 July 2026
A proof that species asymmetry, the forbidden Y-Y channel, and frequency mismatch are each independently necessary for non-trivial hierarchy formation in the model. Introduces an asymmetry index with a proved critical threshold below which no bonds form.
Revised preprint
10 July 2026
Constructs one stochastic DSCD dark sector coupled to standard flat FLRW/GR. Density, pressure, expansion, and BAO distances emerge from the simulated trajectory rather than a fitted w(z) ansatz. The version-1 audit finds individual DSCD parameters unidentifiable from compressed BAO (NO_FORECAST for any single frozen configuration). The version-2 layer then weights 2048 prior-drawn realizations by the joint DR1+DR2 history, passes all predictive-convergence and coverage-calibration gates, and seals thirteen falsifiable DR3 credible intervals.
Cosmological System Audit
10 July 2026
Dataset: Official DESI DR1 and DR2 compressed BAO with full covariance
A coupled stochastic DSCD+GR system evolves asymmetric activity, beat phase, structural memory, regimes, transport, free-capacity density, pressure, and FLRW expansion together. No CAS_LOG, CAS_POLY, CPL, or constant-w function defines the DSCD trajectory.
V1–V5 verify hand-calculated funded accounting, stochastic event hazards, conservative transport, minimal GR closure, and synthetic distance recovery before any real DESI values are used.
With all DSCD dynamics frozen and only θ = H₀r_d/c and Ω_m fitted, χ² = 12.736 on DR1 and 10.280 on DR2. The retrospective DR1 high-redshift conditional score is 3.466, versus 3.492 for ΛCDM.
The depletion combination has condition number 1.06×10⁷ and is not identifiable. Zero-beat, no-memory, no-regime, no-transport, and symmetric-depletion ablations are indistinguishable at current precision. The v1 audit returns NO_FORECAST for any single frozen configuration.
Treating the present DSCD state as latent, 2048 Sobol-drawn realizations weighted by the joint DR1+DR2 likelihood converge predictively: joint effective sample size 205, medians stable under sample halving, disjoint seeds, and DR2-only conditioning, with end-to-end synthetic coverage of 94.9% (95% level) and 73.7% (68% level). The audit returns FORECAST_ELIGIBLE and a DR3 forecast is sealed.
The conditional trajectory relation wC ≥ −1 follows only when the coupled realization has positive density and non-negative depletion. It is a background corollary of the declared GR closure, not a standalone theorem transferred from Part I, and it does not prove the null energy condition or microscopic stability.
Full source hashes, validation artifacts, retrospective residuals, ablations, and audit gates are available in the repository.
One audited stochastic trajectory. The expansion is nearly degenerate with ΛCDM while the DSCD density and interval equation-of-state diagnostic emerge from funded dynamics.
Retrospective full-covariance background fits. These are descriptive goodness-of-fit values, not prospective validation or model probabilities.
Forecast Disposition
10 July 2026
dscd-forecast-v2 · 15/15 gates passed (9 technical, 6 scientific: G1–G4) · Historical records preserved
The present DSCD state is latent. A declared eight-dimensional prior over realizations (depletion scale spanning four orders of magnitude, interaction structure, beat asymmetry, initial activity, matter fraction) is sampled with a scrambled Sobol sequence. Each of 2048 complete DSCD+GR realizations is weighted by its joint DR1+DR2 likelihood with the common BAO scale marginalized analytically. Nothing is fitted to unreleased data.
Joint effective sample size 205. Interval medians shift by less than 0.03 of the 68% width under Sobol halving and disjoint stochastic seed banks, and by 0.064 under DR2-only conditioning. An end-to-end synthetic calibration over twelve prior-drawn truths covers 94.9% at the 95% level and 73.7% at the 68% level.
Thirteen sealed credible intervals on the DR3 tracer layout with 68% widths between 0.16 and 1.10 DR2 sigma. Medians lie within 0.06 sigma of best-fit ΛCDM: the history-compatible DSCD ensemble predicts a ΛCDM-like DR3 and excludes the phantom excursions preferred by unconstrained CPL fits.
The 21 June constant-w surrogate record and the v1 NO_FORECAST disposition are preserved byte-for-byte. Version 2 supersedes them with a hash-chained audit; any post-DR3 change of prior, seed, or thresholds voids the seal.
| Observable | Median | 68% interval | 95% interval |
|---|---|---|---|
| BGS DV/rd, z = 0.295 | 7.901 | 7.865 – 7.939 | 7.830 – 7.985 |
| LRG1 DM/rd, z = 0.510 | 13.253 | 13.199 – 13.311 | 13.146 – 13.378 |
| LRG1 DH/rd, z = 0.510 | 22.451 | 22.388 – 22.510 | 22.276 – 22.581 |
| LRG2 DM/rd, z = 0.706 | 17.408 | 17.346 – 17.471 | 17.284 – 17.544 |
| LRG2 DH/rd, z = 0.706 | 19.989 | 19.927 – 20.044 | 19.808 – 20.101 |
| LRG3+ELG1 DM/rd, z = 0.934 | 21.670 | 21.605 – 21.737 | 21.535 – 21.811 |
| LRG3+ELG1 DH/rd, z = 0.934 | 17.469 | 17.403 – 17.531 | 17.319 – 17.592 |
| ELG2 DM/rd, z = 1.321 | 27.731 | 27.667 – 27.800 | 27.573 – 27.878 |
| ELG2 DH/rd, z = 1.321 | 14.039 | 13.971 – 14.103 | 13.903 – 14.168 |
| QSO DM/rd, z = 1.484 | 29.920 | 29.862 – 29.980 | 29.755 – 30.073 |
| QSO DH/rd, z = 1.484 | 12.873 | 12.806 – 12.937 | 12.742 – 12.999 |
| Lya DH/rd, z = 2.330 | 8.642 | 8.588 – 8.699 | 8.538 – 8.758 |
| Lya DM/rd, z = 2.330 | 38.845 | 38.740 – 38.951 | 38.628 – 39.067 |
Sealed 10 July 2026, before any public DESI DR3 BAO results were available or used. The forecast is judged against the sealed intervals and protocol recorded in the manuscript and hash chain: a DR3 departure below the intervals falsifies the non-phantom realization family, and a departure above the intervals falsifies the calibrated depletion realization family used here.
The sealed forecast as offsets from DR2 central values in DR2 sigma, with best-fit ΛCDM overlaid. Convergence, not coincidence: 2048 structurally different realizations funnel into these intervals.
End-to-end synthetic coverage calibration: twelve prior-drawn truths processed by the identical pipeline. Overall coverage 94.9% (95% level) and 73.7% (68% level).
Keywords
Interacting particle systems · Absorbing-state phase transitions · Stochastic cascades · Irreversible dynamics · Discrete dynamical systems · Landau phase transition · Species asymmetry · Symmetry breaking · Structural hierarchy · Necessity proofs